Dynamic Programming - Incorporating Game Theory in Portfolio Optimization...

MODERN PORTFOLIO THEORY IS DEAD...

In the wake of the financial crisis, a lot of "market experts" were pondering the death of Modern Portfolio Theory... it was thought that the benefits of diversification failed to adequately protect investors from the market rout of 2008 and 2009.  However, as was written in a 2012 piece by John Liu (then of Spartus Capital) titled 'Risk Parity or Risk Parody?' faulty applications of the theory were more likely the cause for the perceived failures of diversification:

"asset allocators unwittingly diversified their 'sources of return' rather than 

their 'sources of risk' as MPT demands.  This failure to properly implement

 MPT perhaps contributed more to losses than the failure of MPT itself."

During the crisis, hedge fund investments performed very well relative to stocks but they have since lagged dramatically as equity markets rallied... powered by seemingly perpetual monetary stimulus.  Consequently, institutions have shunned the alternative asset class over the past few years with notables such as CalPERS pulling out of hedge funds all together.  Factors like high fees, lack of liquidity and transparency and increased competition from mutual funds and ETFs have compounded the larger issue of poor relative performance.

But is now the right time to run away?  

As investors overweight traditional investments, it is entirely possible that the lessons of 2008 have already been forgotten.  Recent market volatility has begun to raise red flags that the stock market party might be over... and if it indeed is, hedge funds could be an important diversification tool to weather a period of lower returns and higher volatility.

VIX DELTA, THE RED FLAG...

I developed a theory that relative changes in the VIX can provide useful insights into future expected equity market performance.

In order to test this hypothesis, I created a simple 3 month-end rolling average of VIX measurements to smooth out some of the variance within the statistic itself.

Over the last five years (through August month end), the S&P has had an annualized return of 13.4% with annualized volatility of about 12%.  During this period, expected volatility as measured by the VIX has been low... the average month-end closing VIX reading during this stretch was under 18.  For comparison, at the end of August it was up to 22.3.  The average monthly change in this measurement during the aforementioned 5 year period was -0.06.  

Why is this important?

It is axiomatic that low volatility is a trademark of bull markets just as high volatility is of bear markets.  However, the VIX can't be viewed without context because a high measurement can easily be associated with rising prices.  Since the VIX is a measurement of expected volatility given current conditions, I have chosen to examine relative changes in the index in relation to future equity market performance.

I looked at 1 year future S&P 500 returns and volatilities and VIX measurements dating back to 1990 to test whether relative VIX movements (or VIX deltas) are better indicators of future performance than absolute measurements.  

Unfortunately, regression fails to find any statistically conclusive evidence that a significant relationship exists with either approach... this wasn't a surprise since equity prices are stochastic.  So in order to draw some sort of conclusion, I resorted to a comparison of the results:

I broke out the results into quintiles according to VIX measurements.  The grid on the left shows the breakdown of S&P 500 performance when using VIX deltas and the grid on the right shows the breakdown when using absolute VIX measurements.

I find the data in the VIX delta grid to be more concise as the standard deviation of returns are more uniform and the average returns for each quintile follow a more logical progression.  I realize this analysis probably wouldn't hold up in a court of law but I feel it objectively supports my thesis nonetheless.

Remember that during the last 5 year period, the S&P had an annualized return of 13.4% with an annualized volatility of 12% and an average VIX delta of -0.06.  I think we can all agree that it was a good run for equities and an especially good run for managers with overweight equity exposure.  Lately, however, we are starting to see a notable shift in VIX delta.  For the last two months of July and August, the VIX deltas have been positive 4.86 and positive 1.35 respectively.  The last time the measurement showed consecutive readings to that magnitude was in the summer of 2011... subsequently, the 1 year S&P future returns and volatilities fell into a range consistent with the fifth quintile of the VIX delta breakdown.

LONG LIVE MODERN PORTFOLIO THEORY...

If we believe that future equity markets will behave consistently with rising VIX deltas then it wouldn't make sense to be overweight the asset class.  So now the question is 'what should equity exposure look like going forward?'

Like I said earlier, hedge funds were a great alternative to equities during financial crisis.  Needless to say, bonds are a ubiquitous asset class that are notably uncorrelated to stocks.  So using equities, hedge funds and bonds we can construct optimal portfolios given our new market expectations.

To represent the behavior of hedge funds and fixed income, I used the Credit Suisse Hedge Fund Index and the Barclays Core US Aggregate Bond Index respectively.  

*Unfortunately, however, I only have data for all three indices dating back to 2004 so my performance metrics will be different from what I used to test my VIX delta hypothesis.

First, let's look at the optimization model using the population of annual index performance.

Below, are grids that represent the expected 1 year performance metrics of each index, the variance-covariance matrix and optimized portfolios under four different scenarios:

The four scenarios I created are as follows:

1. Equal weight each index
2. Maximize portfolio return at 5% annual volatility
3. Maximize portfolio return at 10% annual volatility
4. Optimize portfolio weights to maximize return-to-risk

The optimized portfolios were modeled with the following constraints:

• No short selling
• No portfolio leverage

Now, let's look at what the optimization model looks like given the expectations of rising VIX deltas. 

Here are the 1 year return and volatility quintiles for the indices by VIX delta:

In order to have a suitable amount of data, I combined the bottom two quintiles to generate the following optimization model:

The optimization uses the same scenarios and constraints as previous model but the outcomes are notably different.

Here are the significant differences that I see:

• Expected returns and return/risk measurements were significantly better for the first scenarios given the population expectations but were actually very similar under the fourth scenario that optimizes the return/risk metric.  In both of these scenarios, portfolios were weighted heavily in favor of fixed income.
• The 10% vol scenario in both instances returned a 0% allocation to fixed income
• Optimal equity allocation was never above 25% under either set of market expectations
• In both sets of expectations, hedge fund allocations are significant

An interesting thing to examine at some point will be to see how the optimization model incorporates equity performance in a falling VIX delta environment... I'll follow up later when the metric shifts back to negative.